Finite Group Databasebeta

← #31796129

Subgroups — #31796129

Order 17536.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578425C211yesmaximal, characteristic, complemented
2C4×#14555520C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5C4×D4×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7C4⋊#11764047C2212yescomplemented
8#17984145—21no—
9#17984143—21no—
10#17984143—21no—
11C4×#11764047C2212yescomplemented
12C2×D4×D137C411yescharacteristic
13C22×C4×D137—21no—
14C4×D548C411yescharacteristic
15D4×Dic137C2211yescharacteristic
16#17984050C2211yescharacteristic
17C4×C22⋊D137—21no—
18#17984048—21no—
19C4⋊C4×D137C411yescharacteristic
20C22⋊C4×D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic
23#14555520—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C23×D137—21no—
27C2×D548C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29D4×D137—22no—
30D4×D137—22no—
31C22×Dic137—21no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137—44no—
36C2×C4×D137C2×C411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C4⋊Dic137C2×C411yescharacteristic
40#14555489—21no—
41C22⋊Dic137—21no—
42#14555487—21no—
43#14555487—21no—
44C4×Dic137D411yescharacteristic
45C4×Dic137D411yescharacteristic
46C4×Dic137C2311yescharacteristic
47C2×D4×C137C2×C411yescharacteristic
48C22×C4×C137—21no—
49C4⋊C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51C42×C137C2×C411yescharacteristic
52#11764047—24no—
53#11764047—42no—
54C22×D137—24no—
55C22×D137—21no—
56C22×D137—42no—
57C22×D137—42no—
58C22×D137C22⋊C411yescharacteristic
59D548—21no—
60D548M1612yescomplemented
61C22⋊D137—24no—
62C2×Dic137—21no—
63C2×Dic137—42no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C4×D137—22no—
68C4×D137—21no—
69C4×D137—21no—
70C4×D137—22no—
71C4×D137—42no—
72C4×D137—42no—
73C4×D137—42no—
74C23×C137—21no—
75D4×C137—21no—
76D4×C137M1612yescomplemented
77C2×C4×C137—21no—
78C2×C4×C137—42no—
79C2×C4×C137C22⋊C411yescharacteristic
80C2×C4×C137C22⋊C411yescharacteristic
81C2×C4×C137C22×C411yescharacteristic
82C22×C137—22no—
83C22×C137—41no—
84C22×C137—41no—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—22no—
89C2×D137—48no—
90C2×D137—41no—
91C2×D137—41no—
92C2×D137#10412yes—
93C4×C137—21no—
94C4×C137—21no—
95C4×C137—41no—
96C4×C137C2×M1611yescharacteristic
97C4×C137C2×M1611yescharacteristic
98Dic137—21no—
99Dic137—21no—
100Dic137—41no—
101Dic137AGL(1,ℤ/8)12yes—
102C2×C137—42no—
103C2×C137#40711yescharacteristic
104C2×C137#41811yescharacteristic
105C2×C137C2×#10411yescharacteristic
106D137—22no—
107D137—42no—
108C137#149011yescharacteristic, complemented, Sylow
109#1490—1371nomaximal, Sylow
110C2×C4×D4—1371no—
111#432—1371no—
112C4×M16—1371no—
113D4⋊C8—1372no—
114D4⋊C8—1372no—
115C22×D4—1371no—
116C23×C4—2741no—
117C2×M16—2741no—
118C4×D4—2741no—
119C4×D4—2741no—
120C4×D4—1371no—
121C4×D4—1371no—
122C4×D4—1371no—
123C4×D4#1058773911yescharacteristic
124C2×C4⋊C4—1371no—
125C2×C22⋊C4—2741no—
126C2×C42—1371no—
127C4⋊C8—1372no—
128C22⋊C8—2741no—
129#97—2741no—
130#97—2741no—
131C4×C8—1372no—
132C24—2741no—
133C2×D4—2742no—
134C2×D4—2742no—
135C2×D4—2741no—
136C2×D4—1371no—
137C2×D4C2×#1058773911yescharacteristic
138C22×C4—21no—
139C22×C4—2741no—
140C22×C4—2741no—
141C22×C4—2741no—
142C22×C4—2741no—
143C22×C4—5484no—
144C22×C4—1371no—
145C22×C4—1371no—
146C22×C4—1371no—
147M16—2744no—
148C2×C8—2742no—
149C2×C8—2742no—
150C4⋊C4—2741no—
151C4⋊C4—1371no—
152C4⋊C4C2×#1058773911yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C22⋊C4—2741no—
157C42—1371no—
158C42—1371no—
159C42—1371no—
160C42C2×#1058773911yescharacteristic
161C23—21no—
162C23—2744no—
163C23—2741no—
164C23—5482no—
165C23—5482no—
166C23—1371no—
167D4—21no—
168D4—2744no—
169D4—2741no—
170D4—1372no—
171D4#1455552012yescomplemented
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—2741no—
177C2×C4—2741no—
178C2×C4—2741no—
179C2×C4—5482no—
180C2×C4—5482no—
181C2×C4—5482no—
182C2×C4—5482no—
183C2×C4—1371no—
184C2×C4—1371no—
185C2×C4—1371no—
186C2×C4C22⋊#1058773911yescharacteristic
187C2×C4C22⋊#1058773911yescharacteristic
188C2×C4C22×#1058773911yescharacteristic
189C8—2744no—
190C8—5482no—
191C22—22no—
192C22—41no—
193C22—41no—
194C22—2741no—
195C22—2741no—
196C22—2742no—
197C22—5488no—
198C22—5481no—
199C22—5481no—
200C22—1372no—
201C22C2×C22⋊#1058773911yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—2741no—
206C4—2741no—
207C4—5481no—
208C4—1372no—
209C4D137⋊M1611yescharacteristic
210C4C2×#1455552011yescharacteristic
211C2—42no—
212C2—2742no—
213C2—5482no—
214C2#2257837411yescharacteristic
215C2#2257839511yescharacteristic
216C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C4
cosocleC4×Dic137C23
hypercenterC4×D4#10587739
hypocenterC137#1490
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578425C2semidirect
2. C4×#14555520C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
7. C4⋊#11764047C22semidirect
11. C4×#11764047C22semidirect
60. D548M16semidirect
76. D4×C137M16semidirect
108. C137#1490semidirect
171. D4#14555520semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225784252
C4×#145555202
#225782512
D4⋊#117640472
C4×D4×D1372
#1490137